A class of sensors, including, for example, position sensors (rotary or linear), or optical interferometers (including Sagnac-Effect gyroscopes, for example), derive a measure of a sensed quantity, such as rotation angle, based on the phase of a periodic envelope of a modulated waveform, shown in FIG. 1, where the modulated waveform is a higher-frequency excitation ‘carrier.’ The phase may be derived by sensing two quadrature components of the demodulated waveform. For analog measured signals of the form,Sxm=A cos θs cos ωet Sym=A sin θs cos ωet′, where ωe is the angular frequency of the excitation signal, the demodulated signals are of the form:Sx=Ax cos θs+Bx Sy=Ay sin θs+By 
The physical meaning of the sensor angle θs depends upon the particular application. For example, in the case of a magnetic rotary sensor (as described in U.S. Pat. No. 6,443,536). Ps sensor magnet poles may be spaced at equal submultiples of an armature revolution such that the relationship of θs to mechanical position θm of the shaft is given by
      θ    s    =                    P        s            2        ⁢                  θ        m            .      Other embodiments of measurements, such as of linear displacement (as described in U.S. Pat. No. 6,307,365) or rotation rate (as described in U.S. Pat. No. 4,342,517), for example, based upon the measurement of a phase angle, may all advantageously benefit from the invention described herein. Sensing by electrical, magnetic, optical or any other means leading to the sensing of a modulation from which phase may be derived are encompassed within the scope of the present invention.
More generally, the quadrature components, Sx and Sy, may be derived from two or more sensor signals, Sk, of known phase relationship, and, thus, may be represented as
                              S          x                =                ⁢                                                            A                xc                            ⁢              cos              ⁢                                                          ⁢                              θ                s                                      +                                          A                xs                            ⁢              sin              ⁢                                                          ⁢                              θ                s                                      +                          B              x                                =                                    ∑              k                        ⁢                                          w                kx                            ⁢                              S                k                                                                                      S          y                =                ⁢                                                            A                yc                            ⁢              cos              ⁢                                                          ⁢                              θ                s                                      +                                          A                ys                            ⁢              sin              ⁢                                                          ⁢                              θ                s                                      +                          B              y                                =                                    ∑              k                        ⁢                                          w                ky                            ⁢                                                S                  k                                .                                                        It is sufficient to sensor two signals (not necessarily orthogonal) that span the 2-dimensional space in which the phase angle to be measured is represented as a polar angle. In the case of rotation sensors, it is convenient to space N magnetic sensors magnet poles at equal submultiples of a revolution, such that:
                              w          kx                =                ⁢                              2            N                    ⁢          cos          ⁢                                    2              ⁢              π              ⁢                                                          ⁢              k                        N                                                                        w            ky                    =                    ⁢                                    2              N                        ⁢            sin            ⁢                                          2                ⁢                π                ⁢                                                                  ⁢                k                            N                                      ,            however the present invention does not limit the choice of basis function set for deriving the sensor phase angle θs.
The sensor angle θs may be estimated by ordinary rectangular-to-polar coordinate conversion, i.e., as tan−1 (Sy/Sx), but only to the extent to which the derived signals Sx and Xy are in quadrature and have equal magnitudes (i.e., if the singular values A1 and A2 of the matrix
      [                                        A            xc                                                A            xs                                                            A            yc                                                A            ys                                ]     are equal) and have zero DC offsets (i.e., the extent to which both By and Bx are zero). Gain imbalance (A1≠A2) and offsets may be caused by out-of-tolerance, or degraded, components, and their variability with time may give rise to inaccurate measurement and erratic performance of control systems based on the sensed quantities. For example, if a control loop is used to the control speed of a motor based on an inaccurate measure of rotary position, a periodic speed error (“tach ripple”) will result.
Consequently, a simple method for detecting sensor errors in a context of quadrature detection is highly desirable so as to enable fault-handling procedures to be taken in cases of failure or degradation of the sort discussed above.